7. Required the geometrical mean proportional between 81 and 6561? (151) For 81 729 729: 6561. But the square roots are also proportional (139); viz. 9: 27: 27: 81, whence 27 × 27 = 9 x 81. Therefore, the mean proportional is the product of the square roots of the two extremes. 8. Required 3 mean proportionals between 81 and 6561 ? (150.) 9. To find 4 geometrical means between 2 and 10. Examples of Fractional Powers and Roots. 1. What is the power of 4096,or the cube root of the square of 4096, or the number answering to 40963 4096 log 3.612360 2 3) 7·224720 log. of the square of 4096. Ans. 256 log. 2-408240 log. of the cube root of that square. In this and similar cases, it is best to take the power, or the roct, of the reciprocal of the proposed fraction, and then the reciprocal of that power, or root, will be the answer : Thus the reciprocal of 98 or of 8 is 100% The log. log. of 1. 100 ...log. 0.008774 4.5 43870 35096 98 00394830 log. of the 4% power of the reciprocal. -19605170 log. of 9131 the required power. 1960517 is found by subtracting the log. 0.039483 from the This however, is exactly the same thing as finding the 3d. root of the 4th. power of 2 (because 75 = 4), and therefore 2:5198 is the number denoted by 23. 99 ?. Other Examples. 1. Required the continued product of, and To? Ans. 000000224034 &c. .................the continued product of 27243, and 5899 Ans. 3...........the continued product of 76141, 01779, and 1999 ? Ans. 0.27077 nearly. 4................the continued product of 1.593, 007655, and 01198? 12.... .........a 3d. proportional to 5977 and ·9777 ? Ans. 400185 &c. Ans. Ans. 13..............a 3d. prop ortional to 9876 and 9786? Ans. 14. Required a 4th. proportional to 07655, 1531, and 15791? 15....... Ans. 31582 ..a 4th. proportional to 3777, 2987, and ⚫09876? Ans. 16. Required a 4th. proportional to 3439, 4553, and 5678? Ans. 17. What is the geometrical mean between 07414 and 7414? Ans. 23445 &c. 18... .......between 117 and 357 ? 1404 Ans. 19. Required 2 geometrical means between 3 and 3000 ? 20. Required the cube root of 9999 Ans. Ans. 046417 nearly. 21. Required a 4th. proportional to the cube roots of 17, 19, and 21? Ans. 22. What is the 100th. root of 10? Ans. 1.02329 nearly. 23. To what power must 10 be raised to produce 700 ? Ans. 24. Required the root of .Ans. 037037 nearly, 25. What is the root of?. GEOMETRY. DEFINITIONS. 1. GEOMETRY is that branch of Mathematics in which are considered the properties of lines, surfaces, and solids ; and may be denominated the science of extension or magnitude, in contradistinction to Arithmetic, which is called the science of number. Extension is distinguished into length, breadth, and thick ness. 2. A line is length without breadth or thickness. 3. The extremities of a line are points. And the intersections of one line with another are also points. 4. A surface is that which has length and breadth only. 5. The bounds of a surface are lines. A body or solid has three dimensions, namely, length, breadth, and thickness. The bounds of a solid are surfaces. 6. A right line, or straight line, is that which lies all in the same direction between its extremities; and is the shortest distance between two points. 7. A plane, or plane superficies, is that in which any two points being taken, the right line between them lies wholly in that plane or superficies. 8. A rectilineal angle is the inclination of two straight lines to one another, which meet in a point called the angular point, as the angle C. A D |